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On the rate of convergence of probabilities of moderate deviations

Published online by Cambridge University Press:  09 April 2009

V. K. Rohatgi
V. K. Rohatgi
Affiliation:
The Catholic University of AmericaWashington, D.C.
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Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 1 , February 1970 , pp. 91 - 94
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Baum, L. E., Katz, M. L., and Read, R. R., ‘Exponential convergence rates for the law of large numbers’, Trans. Amer. Math. Soc., 102 (1962), 187–199.Google Scholar
[2]Davis, James Avery, ‘Convergence rates for probabilities of moderate deviations’, Department of Mathematics and Statistics. The University of New Mexico, Tech. Report No. 147, (1967).Google Scholar
[3]Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. (Wiley, New York, 1966).Google Scholar
[4]Heyde, C. C., and Rohatgi, V. K., ‘A pair of domplementary theorems on convergence rates in the law of large numbers’, Proc. Cambr. Phil. Soc., 63 (1967), 73–82.Google Scholar
[5]Loève, M., Probability Theory. (Van Nostrand, 3rd Ed., New York, 1963).Google Scholar
[6]Rubin, H., and Sethuraman, J., ‘Probabilities of moderate deviations’, Sankhya, 27 (1965) Series A, 325–346.Google Scholar